If the roots of the equation `x^3 -9x^2 + 23x-15=0` are in A.P then common difference of that A.P is

Options

A. 3
B. 1
C. 2
D. 4

Answer

To solve the problem, we need to find the common difference of the arithmetic progression (A.P.) formed by the roots of the cubic equation \(x^3 - 9x^2 + 23x - 15 = 0\).

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given cubic equation is \(x^3 - 9x^2 + 23x - 15 = 0\). Here, we can identify:
- \(a = 1\) (coefficient of \(x^3\))
- \(b = -9\) (coefficient of \(x^2\))
- \(c = 23\) (coefficient of \(x\))
- \(d = -15\) (constant term)

2. **Roots in A.P.**:
Since the roots are in A.P., we can denote them as:
- First root: \(a - d\)
- Second root: \(a\)
- Third root: \(a + d\)

Here, \(a\) is the middle term and \(d\) is the common difference.

3. **Sum of the roots**:
According to Vieta's formulas, the sum of the roots of the polynomial is given by:
\[ \text{Sum of roots} = -\frac{b}{a} = -\frac{-9}{1} = 9 \]
Therefore, we have:
\[ (a - d) + a + (a + d) = 9 \]
Simplifying this, we get:
\[ 3a = 9 \implies a = 3 \]

4. **Product of the roots**:
Again, using Vieta's formulas, the product of the roots is given by:
\[ \text{Product of roots} = -\frac{d}{a} = -\frac{-15}{1} = 15 \]
Thus, we have:
\[ (a - d) \cdot a \cdot (a + d) = 15 \]
Substituting \(a = 3\):
\[ (3 - d) \cdot 3 \cdot (3 + d) = 15 \]
This simplifies to:
\[ 3((3 - d)(3 + d)) = 15 \]
\[ 3(9 - d^2) = 15 \]
Dividing both sides by 3:
\[ 9 - d^2 = 5 \]
Rearranging gives:
\[ d^2 = 9 - 5 = 4 \]
Therefore:
\[ d = \pm 2 \]
We will consider the positive value for the common difference, so \(d = 2\).

5. **Roots of the equation**:
Now we can find the roots:
- First root: \(3 - 2 = 1\)
- Second root: \(3\)
- Third root: \(3 + 2 = 5\)

Thus, the roots of the equation are \(1, 3, 5\) and the common difference of the A.P. is \(2\).

### Final Answer:
The common difference of the A.P. is \(2\).